Suppose $X/\mathbb{Q}$ is a reasonable smooth projective variety with interesting etale cohomology. For example, we can say $X$ is an elliptic curve. To what extent does it make sense to study the etale cohomology of $X$ via Berkovich or adic spaces? (I'm asking as a total outsider to the field.) For example, does it make sense to try to learn something about $L(E,s)$ with these tools?

I read the introduction to Berkovich's 1993 IHES paper, but I didn't find any answers there. To what extent do Berkovich spaces allow you to say something about the Galois representation on the Tate module of $E$?